We identify a parameterized family of K 3 surfaces with generic Picard number 19, and we employ elementary methods to determine their local zeta functions. In addition, we explicitly determine those surfaces which are modular.
Let b (n) denote the number of -regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b 4 (n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b 13 (n). In this paper we prove similar families of congruences for b (n) for other values of .
We give exact criteria for the -divisibility of the -regular partition function b (n) for ∈ {5, 7, 11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∞ m=1 (1 − q m ) −1 .
Let b (n) denote the number of -regular partitions of n, where is prime and 3 ≤ ≤ 23. In this paper we prove results on the distribution of b (n) modulo m for any odd integer m > 1 with 3 m if = 3.Recently Ono [7] made a remarkable breakthrough by showing that congruences for p(n) are widespread, proving that for every prime m ≥ 5, there exist infinitely many non-nested arithmetic progressions An + B such that for every nonnegative integer n, p(An + B) ≡ 0 (mod m). Ahlgren [1] has extended this result to include every modulus m coprime to 6, and Ahlgren and Ono gave a unified approach to the problem of such congruences in [3].Given these congruences one may wonder, given an integer m > 1, exactly how often p(n) is divisible by m. While the results mentioned above show thatfor any m coprime to 6, the answer to this question is not known for any m > 1. 295 Int. J. Number Theory 2008.04:295-302. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/02/15. For personal use only. 296 D. Penniston Int. J. Number Theory 2008.04:295-302. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/02/15. For personal use only.
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